M B –O...E.M. acknowledges funding by a two-year ETH Fellowship during 2010–2011. Financial...
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Molecular structure calculationswithout the Born–Oppenheimer approximation
Edit Mátyus1,2 and Markus Reiher2
1 Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, Eötvös University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
2 Laboratory of Physical Chemistry, ETH Zürich, Wolfgang-Pauli-Strasse 10, CH-8093 Zürich, Switzerland
E-mail: [email protected], [email protected]
Introduction
Background
•Most of the present-day theoretical and computational chemistry is basedon the clamped nuclei or Born–Oppenheimer (BO) paradigm.
• This work is devoted to a theoretical approach which does not rely on thisparadigm.
Questions
• Can we reconstruct fundamental chemical concepts in a non-BO theory?
• Can we calculate accurate energy levels and wave functions using thepresent-day computational resources for the ground and excited states ofmolecules with various quantum numbers of the non-relativistic theory?
Starting point
• Our goal here is the development of a variational procedure for the accuratesolution of the Schrödinger equation of few-particle systems.
•We describe electrons and atomic nuclei (or other particles) on equal footing,
Non-Born–Oppenheimer (NBO) theory
Non-relativistic quantum mechanics for n+1 particles
• Schrödinger equation with parameters mi and qi
H = −n+1∑
i=1
12mi
∆r i +n+1∑
i=1
n+1∑
j>i
qiqj
|r i − r j |
• Pauli principle for fermions and bosons with spins si
⇒ 3(n+1) physical parameters: mass, electric charge, and spin
Invariance properties and quantum numbers
• Translational invariance⇒ Translationally invariant coordinates or translationally invariant parame-terization
• Rotation-inversion invariance⇒ Rotational or orbital angular momentum, L,ML, and parity, p
• Permutation invariance of identical particles+ Pauli principle⇒ spin angular momentum S,MS
NBO algorithm: a multi-stage variational pro-cedure
1. Coordinates
• Translationally invariant or laboratory-Vxed Cartesian coordinates, r
2. Quantum Hamiltonian
• Expression in terms of (transl. inv., rectilinear) Cartesian coordinates
3. “Well-parameterized”, symmetry-adapted basis functions
ΦLMLpSMS(r ,σ; A, u, K ,θ) = A{φLMLp(r ; A, u, K ) χSMS
(σ;θ)}
with A = N−1/2p
∑Npi=1 εiPi.
• Eigenfunctions of L2, Lz, p, S2, and Sz
• Gaussian geminals with polynomial prefactors
• Global vector representation for the angular motion
φLMLp(r ;α, u, K ) = |v |2K +LYLML(v ) exp
−12
n+1∑
i=1
n+1∑
j>i
αij(r i − r j)2
, for p = (−1)L
with v =∑n
i=1 uir i ∈ R3, v = v/|v |, K ∈ N0.
4. Overlap and Hamiltonian matrix
• Analytic expressions
• Numerically stable implementation for large values of the polynomial ex-ponents (2K + L > 4)
5. Eigensolver
• Linear combination of (n + 1)-particle basis functions
ΨLMLpSMS(r ,σ; c,α, u, K ,θ) =
Nb∑
I=1
cI ΦLMLpSMS(r ,σ;αI, uI, KI,θI)
• Generalized eigensolver (Intel MKL) for < 1 500×1 500 symmetric matrices
• The number of basis functions, Nb, is increased until convergence
5+1. On-the-Wy optimization of the basis function parameters
• Non-linear parameters, αI, uI, KI, ..., are optimized when Vrst generated andreoptimized regularly during the calculation
• Stochastic variational procedure with a sampling-importance-resamplingstrategy
NBO computer program
A computer program was developed using the FORTRAN language
• Input parameters: mi , qi , si (i = 1, 2, ... , n + 1)
• Eigenstates corresponding to diUerent quantum numbers are obtained inseparate runs (L = 0, 1, 2, ..., ML = −L, ... , +L, p = +1,−1, and S, MS)
• Physical properties are calculated using the wave function
Molecular structure
Classical concepts of molecular structure vs. holistic nature of quantummechanics
• recognize elements of the classical molecular structure using few-particledensity functions
“1-(pseudo)particle” density⇒ radial density
D(1)P,a(R1) = 〈Ψ(r)|δ(ra − rP − R1)|Ψ(r)〉
ρP,a(R) = D(1)P,ab(R1) with R1 = (0, 0, R)
×R
“2-(pseudo)particle” density⇒ angular density
D(2)P,ab(R1, R2) = 〈Ψ(r)|δ(ra − rP − R1)δ(rb − rP − R2)|Ψ(r )〉
ΓP,ab(α) =∫ ∞
0dR1R2
1
∫ ∞
0dR2R2
2 D(2)P,ab(R1, R2) with RT
1R2 = R1R2 cosα
× α
On the emergence of molecular structure
Mass-scale similarity of the Coulomb Hamiltonian
Scaling the masses in the non-relativistic Coulomb Hamiltonian
H(ma, mb, ... , r) → H(ηma, ηmb, ... , r) = ηH(ma, mb, ... , ηr)
is equivalent to scaling the energy and the length
E → ηE and ψ(r) → ψ(ηr)
Radial density vs. mass ratio (ma/mb) in {a±, a±, b∓}-type systems (g.s.,L = 0)
mamb
: 0.000 543 0.5 1 2 5 1840
H− Ps− H+2
{e−, e−, p+} {e−, e−, e+} {p+, p+, e−}
⇒ atomic vs. molecular structure ∼ mass-polarization eUects
• transition point: 0.5 < ma/mb < 1.0
• “alchemical transformation” with respect to the mass ratio
• Ps−: molecular-type system
• H+2: “zero-point vibrations”, ball-like shape for L = 0
• classical vs. quantum structure?
EM, J. Hutter, U. Müller-Herold, and M. Reiher,
Phys. Rev. A 83, 052512 (2011); J. Chem. Phys. 135, 204302 (2011).
Extracting elements of molecular structurefrom the all-particle wave function
Radial and angular densities vs. mass ratio (ma/mb)
in {a±, a±, b∓}- and {a±, a±, b∓, b∓}-type systems (g.s., L = 0)
{e−, e−, p+} {e−, e−, e+} {p+, p+, e−} {e−, e−, p+, p+} {e−, e−, e+, e+} {p+, p+, e−, e−}
ma/mb 0.000 543 1 1 840 0.000 543 1 1 840
ρ0,a
ρ0,b
Γ0,aa
Γ0,bb
⇒ molecular structure ∼ strong correlation eUects for the nuclei
Radial and angular density, H2D+ = {e−, e−, p+, p+, d+} (g.s., L = 0)
ρ0,e ρ0,p ρ0,d ρd,p
Γ0,ee′ Γ0,pp′ Γ0,dp Γd,pp′
SutcliUe and Woolley; CaVero and Adamowicz:
or〈α〉 = (0o+ 180o+ 0o)/3 = 60o 〈α′〉 = (60o+ 60o+ 60o)/3 = 60o
EM, J. Hutter, U. Müller-Herold, and M. Reiher,
Phys. Rev. A 83, 052512 (2011); J. Chem. Phys. 135, 204302 (2011).
Rotation-vibration-electronic excited states
H+2 = {p+, p+, e−}
−0.60
−0.59
−0.58
−0.57
−0.60
−0.59
−0.58
−0.57
−0.49975
−0.49970
−0.49975
−0.49970
Energy[E
h]
A 2Σ+u
−0.49975
−0.49970
Energy[E
h]
A 2Σ+u
X 2Σ+g
(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)
(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)
X 2Σ+g
(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)
(0, 1,0)(1,-1,1)(2, 1,0)(3,-1,1)(4, 1,0)(5,-1,1)
(0, 1,1)(1,-1,0)(2, 1,1)
(L, p, Sp)H (n = 1)
(0, 1,1)(1,-1,0)(2, 1,1)
(L, p, Sp)H (n = 1)
x
x
↑
ortho
even L: A 2Σ
+u
odd L: X 2Σ
+g
x
y
↑
para
even L: X 2Σ
+g
odd L: A 2Σ
+u
Calculated energy levels and dissociation energies of H+2 (examples)
L p Sp E / Eh D / cm−1 Assignment
2 1 1 −0.499 731 516(7) 0.807(1) A 2Σ
+u , v = 0; ortho
...0 1 0 −0.597 139 060(4) 21 379.290(2) X 2
Σ+g , v = 0; para
Refs. Karr and Hilico (2006); Moss (1993)
Ps2 = {e+, e+, e−, e−}
Calculated energy levels of Ps2
L p S+ S− E / Eh η δE / µEh Ref. Assignment
0 1 0 0 −0.516 003 788 7 3.8 · 10−9 −0.001 7 [1] 0+ A1
0 1 1 0 −0.330 287 496 4 3.3 · 10−8 +10.686 [2] 0+ E1 −1 0 0 −0.334 408 295 3 4.8 · 10−8 −0.000 2 [3] 1− B2
Refs. [1] Bubin and Adamowicz (2006); [2] Suzuki and Usukura (2000); [3] Bubin andAdamowicz (2008)
H2 = {p+, p+, e−, e−}
-1.18
-1.08
-0.98
-1.18
-1.08
-0.98
-1.18
-1.08
-0.98
-0.75
-0.65
-0.55
-0.75
-0.65
-0.55
-0.75
-0.65
-0.55
Energy
[Eh] -0.75
-0.65
-0.55
Energy
[Eh]
X 1Σ+g
b 3Σ+u H(n = 1)+H(n = 1)
X 1Σ+g
b 3Σ+u H(n = 1)+H(n = 1)
(0, 1,0,0)(1,-1,1,0)(2, 1,0,0)(3,-1,1,0)(4, 1,0,0)
(0, 1,1,1), (1,-1,0,1)
B 1Σ+u
a 3Σ+g
H(n = 1)+H(n = 2)
(0, 1,1,0)(1,-1,0,0)
(0, 1,0,1)(1,-1,1,1)
(L, p, Sp, Se)
B 1Σ+u
a 3Σ+g
H(n = 1)+H(n = 2)
(0, 1,1,0)(1,-1,0,0)
(0, 1,0,1)(1,-1,1,1)
(L, p, Sp, Se)
x
y
↑
↓singlet, para
even L: X 1Σ
+g
odd L: B 1Σ
+u
x
x
↑
↑triplet, ortho
even L: b 3Σ
+u
odd L: a 3Σ
+g
x
x
↑
↓singlet, ortho
even L: B 1Σ
+u
odd L: X 1Σ
+g
x
y
↑
↑triplet, para
even L: a 3Σ
+g
odd L: b 3Σ
+u
Calculated energy levels of H2
L p Sp Se E / Eh η δE / µEh Ref. Assignment
0 1 0 0 −1.164 025 026 1.4 · 10−8 −0.004 [4] para singlet X 1Σ
+g
1 −1 1 0 −1.163 485 167 3.2 · 10−9 −0.006 [5] ortho singlet X 1Σ
+g
2 1 0 0 −1.162 410 402 2.2 · 10−8 −0.007 [5] para singlet X 1Σ
+g
3 −1 1 0 −1.160 810 486 7.9 · 10−9 −0.006 [5] ortho singlet X 1Σ
+g
4 1 0 0 −1.158 699 660 6.7 · 10−9 −0.006 [5] para singlet X 1Σ
+g
[...]
0 1 1 1 [−0.999 449] [3.9 · 10−5] [−6.6] [6] ortho triplet b 3Σ
+u
1 −1 0 1 [−0.999 444] [3.5 · 10−5] [−11.7] [6] para triplet b 3Σ
+u
[...]
0 1 1 0 −0.753 026 938 1.8 · 10−6 −0.455 [7] ortho singlet B 1Σ
+u
1 −1 0 0 −0.752 848 338 5.9 · 10−6 −2.041 [7] para singlet B 1Σ
+u
[...]
0 1 0 1 −0.730 825 001 1.2 · 10−6 −0.196 [8] para triplet a 3Σ
+g
1 −1 1 1 −0.730 521 131 8.2 · 10−7 −0.279 [8] ortho triplet a 3Σ
+g
[...]
Refs. [4] Bubin and Adamowicz (2003); [5] Pachucki and Komasa (2009); [6] Kołos and Rych-lewski (1990); [7] Wolniewicz, Orlikowski, and Staszewska (2006); [8] Wolniewicz (2007)
7Li = {7Li3+, e−, e−, e−}
Calculated energy levels of 7Li
L p Se E η δE / µEh Ref.
0 1 1/2 −7.477 451 901 8.5 · 10−10 −0.029 [9]1 −1 1/2 −7.409 557 348 9.4 · 10−9
2 1 1/2 −7.334 926 959 7.5 · 10−9 −0.347 [10]
Refs. [9] Stanke, Kedziera, Bubin, and Adamowicz (2007);[10] Sharkley, Bubin, and Adamowicz (2011)
EM and M. Reiher, J. Chem. Phys. to be submitted (2012).
Acknowledgement
E.M. acknowledges funding by a two-year ETH Fellowship during 2010–2011.Financial support from the Swiss National Fund (SNF) is also gratefully ac-knowledged. E.M. also thanks the Hungarian ScientiVc Research Fund (OTKA,NK83583) and the European Union and the European Social Fund (TÁMOP-4.2.1/B-09/1/KMR-2010-0003) for Vnancial support during 2012.